where \(k(t)\)  is the strength of the linear coupling among the two fields and \(k^{*}\)  denotes
the complex conjugate of \(k\) . The time-modulated strength of the balanced loss gain terms is \(\gamma (t)\) .
The space-time modulated strengths of the non-linear interaction are denoted as \(g_{ij}(x,t)\) .
In the terminology of optics, \(g_{11}, g_{22}\)  are self phase modulation terms, while \(g_{21},g_{12}\) 
correspond to cross-phase modulation. The external potential \(V(x)\)  is relevant
for the mean-field description of Bose-Einstein condensates. Several \({\cal {PT}}\) -symmetric complex potentials
\(V(x)\)  exhibit many interesting phenomenon. We discuss both real and complex potentials separately in this article
and choose \(V(x)=s(x)+i\tilde{s}(x)\) , where functions \(s(x), \tilde{s}(x)\)  are real.
There are several solvable limits of Eq.(REF ), starting from the simplest case of
\(g_{ij} = 0, k = \gamma =0\)  which models the optical wave propagation under paraxial approximation[1]}, [2]}.
The integrable canonical form of Manakaov-Zakharov-Schulman(MZS)
system[3]}, [4]} is obtained for constant coefficients \(g_{ij}\)  with \(g_{11}=g_{21}, g_{12}=g_{22}\)  and vanishing
linear coupling, loss-gain terms and external potential.
Exact solutions of MZS system with non-vanishing constant loss-gain terms and time-dependent \(g_{ij}\)  with specific
time-modulation have been constructed via non-unitary transformations[5]}. The solutions for constant \(g_{ij}\)  appears
as a special case. The Gross-Pitaevskii equation, which provides a mean-field description of BEC, is obtained for vanishing
loss-gain terms and non-vanishing \(V(x)\) . Several exact solutions for different choices of \(V(x)\)  and space-modulated
nonlinear strengths have been constructed[6]}. It appears that solvable limits of Eq. (REF ) in its generic form,
particularly with time-dependent \(k, \gamma \) , space-time dependent \(g_{ij}\)  and with or without the external potential \(V(x)\) ,
have not been investigated so far. The method of non-unitary transformation[5]} will be employed to investigate
solvability of Eq. (REF ) in its generic form.
